Existence of Solution for A Coupled System of Delay Functional Differential Equations

Abstract

This study addresses the existence of solutions for a coupled system of delay functional differential equations, a class of models that arises naturally in phenomena where the current rate of change depends not only on the present state but also on past states. Such systems are widely used in applications including population dynamics, control theory, and engineering processes with memory effects. The analysis is carried out within an appropriate functional framework, where the problem is reformulated as an equivalent integral equation. By employing fixed-point theorems such as the Banach contraction principle and Schauder fixed-point theorem, sufficient conditions for the existence (and in some cases uniqueness) of solutions are established. The study considers both continuous and bounded delay functions and imposes suitable assumptions on the nonlinear terms, including continuity, boundedness, and Lipschitz-type conditions. Furthermore, the work examines the influence of delay parameters on the behavior of the system and discusses the stability of solutions under certain constraints. The theoretical results are supported by illustrative examples that demonstrate the applicability of the derived conditions. Overall, this paper contributes to the qualitative theory of delay differential equations by providing rigorous existence results for coupled systems, thereby offering a foundation for further analytical and numerical investigations in applied sciences.